volume of solid of revolution
Let us consider a solid of revolution, which is generated when a planar domain D rotates about a line of the same plane. We chose this line for the x-axis, and for simplicity we assume that the boundaries of D are the mentioned axis, two ordinates x=a, x=b(>a), and a continuous curve y=f(x).
Between the bounds a anb b we fit a sequence of points x1,x2,…,xn-1 and draw through these the ordinates which divide the domain D in n parts. Moreover we form for every part the (maximal) inscribed and the (minimal) circumscribed
rectangle
. In the revolution of D, each rectangle generates a circular cylinder. The considered solid of revolution is part of the volume V> of the union of the cyliders generated by the circumscribed rectangles and at the same time contains the volume V< of the union of the cylinders generated by the inscribed rectangles.
Now it is apparent that
V>=π[M21(x1-a)+M22(x2-x1)+…+M2n(b-xn-1)], |
V<=π[m21(x1-a)+m22(x2-x1)+…+m2n(b-xn-1)], |
where M1,M2,…,Mn are the greatest and m1,m2,…,mn the least values of the continuous function f on the intervals (http://planetmath.org/Interval) [a,x1], [x1,x2], …, [xn-1,b]. The volume V of the solid of revolution thus satisfies
V<≤V≤V>, |
and this is true for any x1<x2<…<xn-1 of the interval [a,b]. The theory of the Riemann integral guarantees that there exists only one real number V having this property and that it is also the definition of the integral ∫baπ[f(x)]2𝑑x. Therefore the volume of the given solid of revolution can be obtained from
V=π∫ba[f(x)]2𝑑x. |
References
- 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title | volume of solid of revolution |
---|---|
Canonical name | VolumeOfSolidOfRevolution |
Date of creation | 2013-03-22 17:20:12 |
Last modified on | 2013-03-22 17:20:12 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Topic |
Classification | msc 51M25 |
Related topic | PappussTheoremForSurfacesOfRevolution |
Related topic | SurfaceOfRevolution |
Related topic | VolumeAsIntegral |